Dr Edward Crane

Biography

My research is in probabilty and complex analysis.

Current projects

My main project in probability concerns mathematical models of forest fires that display self-organized criticality. These are not intended to be realistic models of real-world forest fires, but instead they give us insight into complex stochastic systems that are driven into critical states by their own dynamics. The models that we study combining simple evolution rules and randomness to obtain this behaviour. The simplicity enables us to carry out exact mathematical analysis. In the critical state, the distribution of sizes of fires has a power law tail, and even the smallest possible change to the driving randomness has a positive probability of propagating into a macroscopic change in the state of the system. Mathematically, this research involves a mixture of probability theory and partial differential equations.

My recent work in complex analysis is about circle packing. The Koebe-Andreev-Thurston theorem says that given any triangulation of the sphere, we can find a packing of discs on the sphere with disjoint interiors, whose graph of tangencies is isomorphic to the given triangulation. There is a growing theory of discrete conformal mapping based on circle packing. My current project (with Ken Stephenson and James Ashe) involves extending this to a good theory of circle packing with movable branch points, in order to investigate the analogy with analytic functions that are not locally univalent. For example, we hope it will help to us to tackle the existence and uniqueness question for the circle packing analogue of rational functions.